A few decades ago I read the transcript of a wonderful lecture by mathematician Mark Kac; it's perhaps available on film or videotape nowadays. The topic was "Can You Hear the Shape of a Drum?" In other words, what information is carried by the overtones of a vibrating membrane?

Amazingly enough, one can "hear" a lot: the total surface area, the circumference, and the number of holes in the drumhead, for starters. (To a mathematician the drum needn't be circular, or even a solid sheet of material.) The various normal modes of oscillation for a thin film under tension somehow encode a host of detail about the object.

Why was this such an exciting idea that a kid would remember it over a quarter century later? Maybe the magical spell came from seeing an unexpected partnership between numbers and the world, a surprise marriage of theory and reality. Applied math, of which Kac was a master, is that sort of enterprise: a game of deep patterns and startling connections. Similar magic comes in other areas of human knowledge, ranging from history to psychology, from economics to anatomy. Suddenly, one sees a why --- a reason for what at first seemed arbitrary --- a meaning behind an apparently random structure. A lovely idea, unveiled. Lightning strikes, sha-zap!